# An Introduction to Differential Manifolds by Jacques Lafontaine

By Jacques Lafontaine

This ebook is an creation to differential manifolds. It provides strong preliminaries for extra complicated themes: Riemannian manifolds, differential topology, Lie idea. It presupposes little heritage: the reader is just anticipated to grasp uncomplicated differential calculus, and a bit point-set topology. The e-book covers the most subject matters of differential geometry: manifolds, tangent house, vector fields, differential varieties, Lie teams, and some extra subtle subject matters equivalent to de Rham cohomology, measure concept and the Gauss-Bonnet theorem for surfaces.

Its ambition is to offer reliable foundations. specifically, the advent of “abstract” notions comparable to manifolds or differential kinds is encouraged through questions and examples from arithmetic or theoretical physics. greater than a hundred and fifty workouts, a few of them effortless and classical, a few others extra subtle, may also help the newbie in addition to the extra specialist reader. strategies are supplied for many of them.

The ebook could be of curiosity to varied readers: undergraduate and graduate scholars for a primary touch to differential manifolds, mathematicians from different fields and physicists who desire to collect a few feeling approximately this gorgeous theory.
The unique French textual content advent aux variétés différentielles has been a best-seller in its type in France for lots of years.

Jacques Lafontaine used to be successively assistant Professor at Paris Diderot collage and Professor on the collage of Montpellier, the place he's almost immediately emeritus. His major examine pursuits are Riemannian and pseudo-Riemannian geometry, together with a few points of mathematical relativity. in addition to his own examine articles, he was once inquisitive about numerous textbooks and examine monographs.

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Additional info for An Introduction to Differential Manifolds

Example text

Since the function θ → θ is Lipschitz (with constant 1) on S n−1 , we know from Lecture 8 that, for any t ≥ 0, σ 2 θ − M > t ≤ 2e−nt /2 . In particular, σ θ − M > M γ ≤ 2e−nM 2 γ 2/2 . So M (1 − γ) ≤ θ ≤ M (1 + γ) 2 2 on all but a proportion 2e−nM γ /2 of the sphere. Let A be a δ-net on the sphere in Rk with at most (4/δ)k elements. Choose a random embedding of Rk in Rn : more precisely, fix a particular copy of Rk in Rn and consider its images under orthogonal transformations U of Rn as a random subspace with respect to the invariant probability on the group of orthogonal transformations.

Talagrand, “A new isoperimetric inequality”, Geom. Funct. Anal. 1:2 (1991), 211–223. AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY 55 [Talagrand 1991b] M. Talagrand, “A new isoperimetric inequality and the concentration of measure phenomenon”, pp. 94–124 in Geometric aspects of functional analysis (Israel Seminar, 1989–1990), edited by J. Lindenstrauss and V. D. Milman, Lecture Notes in Math. 1469, Springer, 1991. [Tomczak-Jaegermann 1988] N. Tomczak-Jaegermann, Banach–Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Math.

In a similar way, a sequence X1 , X2 , . . , Xn of independent random variables arises if each variable is defined on the product space Ω1 × Ω2 × . . × Ωn and Xi depends only upon the i-th coordinate. The second crucial idea, which we will not discuss in any depth, is the use of many different σ-fields on the same space. The simplest example has already been touched upon. The product space Ω1 ×Ω2 carries two σ-fields, much smaller than the product field, which it inherits from Ω1 and Ω2 respectively.